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law of averages

Contributor(s): Stan Gibilisco

The law of averages is an erroneous generalization of the law of large numbers, which states that the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. The law of averages is usually mentioned in reference to situations without enough outcomes to bring the law of large numbers into effect.  

A common example of how the law of averages can mislead involves the tossing of a fair coin (a coin equally likely to come up heads or tails on any given toss). If someone tosses a fair coin and gets several heads in a row, that person might think that the next toss is more likely to come up tails than heads in order to "even things out." But the true probabilities of the two outcomes are still equal for the next coin toss and any coin toss that might follow. Past results have no effect whatsoever: Each toss is an independent event.

Another example of the law of averages involves batting averages in baseball. If a player has a batting average of .250, then he can be expected to get a hit on one out of every four at-bats (not counting bases on balls) in the long term. However, as anyone who follows baseball knows, hitters' fortunes run in "streaks" and "slumps" that can last for days or even weeks. During a "streak," a batter might get a hit in four out of 10 at-bats, and during "slumps" he might get a hit in only one out of 10 at-bats. If people invoke the law of averages when the hitter is "slumping," they will say that "he is due for a hit," suggesting on each and every at-bat that his chances are better than one in four because things "have to even out." However, according to the strict law of large numbers, no such supposition can be made.

The law of large numbers is often confused with the law of averages, and many texts use the two terms interchangeably. However, the law of averages, strictly defined, is not a law at all, but a logic error that is sometimes referred to as the gambler’s fallacy.

This was last updated in December 2012

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